Chapter 8.7, Problem 24E

To determine

To find: The binomial series to expand the function as the power series and state the radius of the convergence.

Answer to Problem 24E

The required binomial series is $1-\frac{2}{3}x+{\displaystyle \sum _{n=2}^{\infty }\frac{1\cdot 4\cdot 7\cdot \cdot \left(3n-5\right)}{3^n\cdot n!}{x}^n}$ with $R=1$ .s

Explanation of Solution

Given:

The given expression is ${\left(1-x\right)}^{\frac{2}{3}}$ .

Calculation:

Consider the given expression is,

  ${\left(1-x\right)}^{\frac{2}{3}}$

Consider the formula for the binomial expansion is,

  $\begin{array}{c}{\left(1+x\right)}^k={\displaystyle \sum _{n=0}^{\infty }\left(\begin{array}{c}k\\ n\end{array}\right)x^n}\\ =1+kx+\frac{k\left(k-1\right)}{2!}{x}^2+\frac{k\left(k-1\right)\left(k-2\right)}{3!}{x}^3\mathrm{...}\end{array}$

Then, the binomial coefficient is,

  ${\left(1-x\right)}^{\frac{2}{3}}$

Consider $\frac{2}{3}$ as the real numbers… $\left|-x\right|<1$

Then,

  $\begin{array}{c}{\left(1-x\right)}^{\frac{2}{3}}={\displaystyle \sum _{n=0}^{\infty }\left(\begin{array}{c}\frac{2}{3}\\ n\end{array}\right)}{\left(-x\right)}^n\\ =1+\frac{2}{3}\left(-x\right)+\frac{\frac{2}{3}\left(-\frac{1}{3}\right)}{2!}{\left(-x\right)}^2+\frac{\frac{2}{3}\left(-\frac{1}{3}\right)\left(-\frac{4}{3}\right)}{3!}{\left(-x\right)}^3++\frac{\frac{2}{3}\left(-\frac{1}{3}\right)\left(-\frac{7}{3}\right)}{4!}{\left(-x\right)}^4+\mathrm{....}\\ =1-\frac{2}{3}x+{\displaystyle \sum _{n=2}^{\infty }\frac{{\left(-1\right)}^{n-1}{\left(-1\right)}^n\cdot 2\cdot \left[1\cdot 4\cdot 7\cdot \cdot \cdot \left(3n-5\right)\right]}{3^n\cdot n!}}{x}^n\\ =1-\frac{2}{3}x+{\displaystyle \sum _{n=2}^{\infty }\frac{1\cdot 4\cdot 7\cdot \cdot \left(3n-5\right)}{3^n\cdot n!}{x}^n}\end{array}$

The above equation is of the interval,

  $\begin{array}{l}\left|-x\right|<1\\ \left|x\right|>1\end{array}$

Thus, $R=1$

Thus, the required binomial series is $1-\frac{2}{3}x+{\displaystyle \sum _{n=2}^{\infty }\frac{1\cdot 4\cdot 7\cdot \cdot \left(3n-5\right)}{3^n\cdot n!}{x}^n}$ with $R=1$ .